Determining the Fine Structure Constant
G. Gabrielse
Department of Physics, Harvard University
gabrielse@physics.harvard.edu
To appear in “Lepton Dipole Moments: The Search for Physics Beyond the Standard
Model”, edited by B.L. Roberts and W.J. Marciano (World Scientific, Singapore, 2009),
Advanced Series on Directions in High Energy Physics – Vol. 20.
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
Chapter 8
Determining the Fine Structure Constant
G. Gabrielse
Department of Physics, Harvard University
17 Oxford Street, Cambridge, MA 02138
gabrielse@physics.harvard.edu
The most accurate determination of the ﬁne structure constant α is
α−1 = 137.035 999 084 (51) [0.37 ppb]. This value is deduced from the
measured electron g/2 (the electron magnetic moment in Bohr magne-
tons) using the relationship of α and g/2 that comes primarily from
Dirac and QED theory. Less accurate by factors of 12 and 21 are deter-
minations of α from combined measurements of the Rydberg constant,
two mass ratios, an optical frequency, and a recoil shift for Rb and Cs
atoms. Helium ﬁne structure intervals have been measured well enough
to determine α with nearly the same precision – if two-electron QED
calculations can be sorted out. Less accurate measurements are also
compared.
Contents
8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
8.2. Importance of the Fine Structure Constant . . . . . . . . . . . . . . . . . . . 265
8.3. Most Accurate α Comes from Electron g/2 . . . . . . . . . . . . . . . . . . . 266
8.3.1. New Harvard Measurement and QED Theory . . . . . . . . . . . . . . 266
8.3.2. Status and Reliability of the QED Theory . . . . . . . . . . . . . . . . 269
8.3.3. How Much Better can α be Determined? . . . . . . . . . . . . . . . . . 273
8.4. Determining α from the Rydberg, Two Mass Ratios and /M for an Atom . 274
8.5. Other Measurements to Determine α . . . . . . . . . . . . . . . . . . . . . . . 278
8.5.1. Determining α from He Fine Structure . . . . . . . . . . . . . . . . . 278
8.5.2. Historically Important Methods . . . . . . . . . . . . . . . . . . . . . . 281
8.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
263
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
264 G. Gabrielse
8.1. Introduction
The fundamental and dimensionless ﬁne structure constant α is deﬁned (in
SI units) by
1 e2
α= . (8.1)
4πǫ0 c
The well known value α−1 ≈ 137 is not predicted within the Standard
Model of particle physics.
The most accurate determination of α comes from a new Harvard mea-
surement [7, 8] of the dimensionless electron magnetic moment, g/2, that
is 15 times more accurate than the measurement that stood for twenty
years [9]. The ﬁne structure constant is obtained from g/2 using the theory
of a Dirac point particle with QED corrections [10–15]. The most accurate
α, and the two most accurate independent values, are given by
α−1 (H08) = 137.035 999 084 (51) [0.37 ppb] (8.2)
α−1 (Rb08) = 137.035 999 45 (62) [4.5 ppb] (8.3)
α−1 (Cs06) = 137.036 000 0 (11) [8.0 ppb]. (8.4)
Fig. 8.1. compares the most accurate values.
ppb
10 5 0 5 10 15
Harvard g 2 2008
UW g 2 1987 Harvard g 2 2006
Rb 2008
Rb 2006
Cs 2006
599.80 599.85 599.90 599.95 600.00 600.05 600.10
1 5
Α 137.03 10
Fig. 8.1. The most precise determinations of α.
The uncertainties in the two independent determinations of α are within
a factor of 12 and 21 of the α from g/2. They rely upon separate mea-
surements of the Rydberg constant [16, 17], mass ratios [18, 19], optical
frequencies [20, 21], and atom recoil [21, 22]. Theory also plays an impor-
tant role for this method, to determine the Rydberg constant (reviewed in
Ref. 23) and one of the mass ratios [24].
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
Determining the Fine Structure Constant 265
In what follows, the importance of the ﬁne structure constant is dis-
cussed ﬁrst. Determining α from the measured electron g/2 comes next,
starting with an operational summary of how this is done, and ﬁnishing
with an overview of the status and reliability of the theory. Determining
α from the combined measurements mentioned above is the next topic.
The possibility to determine α with nearly the same precision from atomic
ﬁne structure is then considered. Helium ﬁne structure intervals have been
measured with enough accuracy to do so, [1–4, 25] if inconsistencies in the
needed two-electron QED theory [5, 6] can be cleared up. Other methods
that are important for historical reasons are mentioned, and followed by a
conclusion.
8.2. Importance of the Fine Structure Constant
The ﬁne structure constant appears in many contexts and is important for
many reasons.
(1) The ﬁne structure constant is the low energy electromagnetic cou-
pling constant, the measure of the strength of the electromagnetic
interaction in the low energy limit.
(2) The ﬁne structure constant is the basic dimensionless constant of
atomic physics, distinguishing the energy scales that are important
for atoms. In terms of the electron rest energy, me c2 :
(a) The binding energy of an atom is approximately α2 me c2 .
(b) The ﬁne structure energy splitting in atoms goes as α4 me c2 .
(c) The hyperﬁne structure energy splitting goes as
(me /M ) α4 me c2 , like the ﬁne structure splitting except re-
duced by an additional ratio of an electron mass to the nucleon
mass (M ).
(d) The lamb shift in an atom goes as α5 me c2 .
(3) The ﬁne structure constant is also important for condensed matter
physics, the condensed matter and atomic energy scales being sim-
ilar. Important examples include the quantum hall resistance and
the oscillation frequency of a Josephson junction.
(4) The ﬁne structure constant is a important and central to our in-
terlinked system of fundamental constants [23]. Its role will be
enhanced if a contemplated redeﬁnition of the SI system of units
(to remove the dependence upon an artifact mass standard) is
adopted [27].
(5) Measurements of the muon magnetic moment [28], made to test
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
266 G. Gabrielse
for possible breakdowns of the Standard Model of particle physics,
require a value for α. Small departures from the Standard Model
would only be visible once the large α-dependent QED contribution
to the muon g value is subtracted out.
(6) Comparing α values from methods that depend diﬀerently upon
QED theory is a test of the QED theory.
8.3. Most Accurate α Comes from Electron g/2
8.3.1. New Harvard Measurement and QED Theory
The most accurate determination of the ﬁne structure constant utilizes
a new measurement of the electron magnetic moment, measured in Bohr
magnetons, [7]
g/2 = 1.001 159 652 180 73 (28) [0.28 ppt]. (8.5)
This 2008 measurement of g/2 (Chapter 6) is 15 times more precise than
the 1987 measurement [9] that had stood for about twenty years. The high
precision and accuracy came from new methods that made it possible to
resolve the quantum cyclotron levels [29], as well as the spin levels, of one
electron suspended for months at a time in a cylindrical Penning trap [30].
The electron g/2 is essentially the ratio of the spin and cyclotron fre-
quencies. This ratio is deduced from measurable oscillation frequencies in
the trap using an invariance theorem [31]. These frequencies are measured
using quantum jump spectroscopy of one-quantum transitions between the
lowest energy levels [8]. The cylindrical Penning trap electrodes form a
microwave cavity that shapes the radiation ﬁeld in which the electron is
located, narrowing resonance linewidths by inhibiting spontaneous emis-
sion [29, 32], and providing boundary conditions which make it possible to
identify the symmetries of cavity radiation modes [7, 33]. A QND (quan-
tum nondemolition) coupling, of the cyclotron and spin energies to the
frequency of an orthogonal and nearly harmonic electron axial oscillation,
reveals the quantum state [29]. This harmonic oscillation of the electron
is self-excited [34], by a feedback signal [35] derived from its own motion,
to produce the large signal-to-noise ratio needed to quickly read out the
quantum state without ambiguity.
Within the Standard Model of particle physics the measured electron
g/2 is related to the ﬁne structure constant by
α α 2 α 3 α 4 α 5
g/2 = 1+ C2 + C4 + C6 + C8 + C10
π π π π π
+ . . . + ahadronic + aweak (8.6)
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
Determining the Fine Structure Constant 267
Dirac theory of the electron provides the leading term on the right. Fig. 8.2.
compares the size of the measured g/2 (gray) with its measurement uncer-
tainty (black) to size of this leading Dirac term and other theoretical con-
tributions (gray). The uncertainties (black) of the theoretical contributions
arise from the uncertainty for the coeﬃcients.
ppt ppb ppm
Harvard g 2
1
C2 Α Π
2
C4 Α Π
3
C6 Α Π
4
C8 Α Π
5
C10 Α Π
hadronic
weak
15 12
10 10 10 9 10 6 10 3 1
contribution to g 2 1 a
Fig. 8.2. Contributions to g/2 for the experiment (top bar), terms in the QED series
(below), and from small distance physics (below). Uncertainties are black. The inset
light gray bars represent the magnitude of the larger mass-independent terms (A1 ) and
the smaller A2 terms that depend upon either me /mµ or me /mτ . The even smaller A3
terms, functions of both mass ratios, are not visible on this scale.
Quantum electrodynamics (QED) provides the expansion in the small
ratio α/π ≈ 2×10−3 , and the values of the coeﬃcients Ck . The ﬁrst three of
these, C2 [10], C4 [11–13], C6 [14] are exactly known functions which have no
theoretical uncertainty. The small uncertainties in C4 and C6 , completely
negligible at the current level of experimental precision (Fig. 8.2.), arise
because C4 and C6 depend slightly upon lepton mass ratios.
C2 = 0.500 000 000 000 00 (exact) (8.7)
C4 = − 0.328 478 444 002 90 (60) (8.8)
C6 = 1.181 234 016 827 (19) (8.9)
C8 = − 1.914 4 (35) (8.10)
C10 = 0.0 (4.6). (8.11)
There is no analytic solution for C8 yet but this coeﬃcient has been calcu-
lated numerically [15]. Unfortunately, C10 has not yet been calculated; the
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
268 G. Gabrielse
quoted bound is a simple extrapolation from the lower-order Ck [36].
Very small additional contributions due to short distance physics have
also been evaluated [37, 38],
ahadronic = 0.000 000 000 001 682 (20) (8.12)
aweak = 0.000 000 000 000 030 (01) (8.13)
The hadronic contribution is important at the current level of experimental
precision, but the reported uncertainty for this contribution is much smaller
than is currently needed to determine α from g/2.
The most precise value of the ﬁne structure constant comes from using
the very accurately measured electron g/2 (Eq. 8.5) in the Standard Model
relationship between g/2 and α (Eq. 8.6). The result is
α−1 (H08) = 137.035 999 084 (33) (39) [0.24 ppb] [0.28 ppb],
= 137.035 999 084 (33) (12) (37) [0.24 ppb] [0.09 ppb] [0.27 ppb],
= 137.035 999 084 (51) [0.37 ppb]. (8.14)
The ﬁrst line shows experimental (ﬁrst) and theoretical (second) uncer-
tainties that are nearly the same. The second line separates the theoretical
uncertainty into two parts, the numerical uncertainty in C8 (second) and
the estimated uncertainty for C10 (third). The third line gives the total
0.37 ppb uncertainty. A graphical comparison of the experimental and
theoretical uncertainties in determining α from g/2 is in Fig. 8.3..
0.4
total uncertainty
uncertainty in Α Α in ppb
0.3 from from theory
exp't
0.2
0.1
0.0
Σg2 Σ C8 Σ C10 Σ ahadronic Σ aweak
Fig. 8.3. Experimental uncertainty (black) and theoretical uncertainties (gray) that
determine the uncertainty in the α that is determined from the measured electron g/2
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
Determining the Fine Structure Constant 269
The crudely estimated theoretical uncertainty in the uncalculated C10
currently adds more to the uncertainty in α more than does the measure-
ment uncertainty for g/2. As a result, the factor of 15 reduction in the
measurement uncertainty for g/2 results in only a factor of 10 reduction in
the uncertainty in α.
Fig. 8.1. compares our α−1 (H08) to other accurate determinations of α.
The ﬁne structure constant is currently determined about 12 and 21 times
more precisely from g/2 than from the best Cs and Rb measurements (to
be discussed). No other α determination has error bars small enough to
ﬁt in this ﬁgure. Comparing our α with the most accurate independent
determinations is a test of the Standard Model prediction in Eq. 8.6, along
with the theoretical assumptions used for the other determinations. More
accurate independent α values would improve upon what is already the
most stringent test of QED theory.
8.3.2. Status and Reliability of the QED Theory
The electron g/2 diﬀers from 1 by about one part in 103 as a result of the
QED corrections to the Dirac theory. How uncertain and how reliable is
the QED theory that is needed to accurately determine α from g/2? Given
the complexity of the theory, and mistakes that have been discovered in
the past, how likely is it that additional mistakes will either appreciably
change α in the future, or go undetected?
In this section we summarize the status of calculations of the Ck coeﬃ-
cients, the current values of which are already listed in Eqs. 8.7-8.11. The
history and method of the calculations are discussed in Chapters 3 and 5.
We illustrate how impressive analytic calculations have made it easy to now
evaluate the lowest order coeﬃcients (C2 , C4 and C6 ) to an arbitrary pre-
cision with no theoretical uncertainty, provided that no mistakes have been
made. Numerical calculations and veriﬁcations of C8 , and the prospects for
numerical calculations of C10 , are also summarized.
There is no theoretical uncertainty in the Dirac unit contribution to
g/2 in Eq. 8.6. There is also no theoretical uncertainty in the leading QED
correction, C2 (α/π), insofar as long ago a single Feynman diagram was
evaluated analytically to determine C2 exactly [10].
The C4 coeﬃcient is the sum of a mass-independent term and two much
smaller terms that are functions of lepton mass ratios,
(4) (4) me (4) me
C4 = A1 + A2 ( ) + A2 ( ). (8.15)
mµ mτ
The mass-independent term is larger by many orders of magnitude. This
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
270 G. Gabrielse
pure number, involving 7 Feynman diagrams, is given by [11–13, 39]
(4) 197 π 2 3 π2
A1 = + + ζ(3) − ln (2) (8.16)
144 12 4 2
= −0.328 478 965 579 193 . . . (8.17)
where ζ(s) is the Riemann zeta function (Zeta[s] in Mathematica).
There is no theoretical uncertainty in this contribution, which can easily
be evaluated to any desired precision. Of course, this is only true if there are
no mistakes in the analytic derivation. The original result [40] had an error
in the evaluation of an integral. This was corrected some years later [12]
(and then conﬁrmed independently [11]) after the initial result did not agree
with a numerical calculation. This was the ﬁrst of several instances where
independent evaluations allowed the elimination of mistakes, as we shall
see.
(4)
The mass-dependent function A2 (x) is an analytical evaluation of one
Feynman diagram [41]. In a convenient form [42] it is given by
(4) 25 ln(x) x
A2 (1/x) = − − + x2 [4 + 3 ln(x)] + (1 − 5x2 )
36 3 2
π2 1−x
× − ln(x) ln( ) − Li2 (x) + Li2 (−x)
2 1+x
π2 1
+ x4 − 2 ln(x) ln( − x) − Li2 (x2 ) . (8.18)
3 x
The dilogarithm function is a special case of the polylogarithm (Poly-
∞
Log[n,x] in Mathematica); it has a series expansion Lin (x) = k=1 xk /kn
that converges for the cases we need. The exactly calculated mass-
dependent function is evaluated as a function of two lepton mass ratios
[23, 43],
mµ /me = 206.768 276 (24) (8.19)
mτ /me = 3 477.48 (57) (8.20)
There is no theoretical uncertainty in the mass-dependent terms
(4)
A2 (me /mµ ) = 5.197 387 71 (12) × 10−7 , (8.21)
(4) −9
A2 (me /mτ ) = 1.837 63 (60) × 10 . (8.22)
The uncertainties are from the uncertainties in the measured mass ratios.
When multiplied by (α/π)2 these are very small contributions to g/2. The
ﬁrst of these two contributions is larger than the current experimental pre-
cision (Fig. 8.2.) while the second is not. The uncertainties in both terms
are so small as to not even be visible in Fig. 8.2..
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
Determining the Fine Structure Constant 271
The higher order coeﬃcients, Ck with k = 6, 8, 10, . . ., are each the sum
of a constant and functions of mass ratios,
(k) (k) me (k) me (k) me me
Ck = A1 + A2 ( ) + A2 ( ) + A3 ( , ). (8.23)
mµ mτ mµ mτ
(k)
The leading mass-independent term, A1 , is much larger than the small
mass-dependent corrections. In fact, for k ≥ 8, the mass-dependent cor-
rections should not be needed to determine α from g/2 at the current or
foreseeable measurement precision in g/2 owing to their very small values.
For sixth order the mass-independent term requires the evaluation of 72
Feynman diagrams. An analytic evaluation of this term, mostly by Remiddi
and Laporta [14], is
(6) 83 2 215 239 4 28259
A1 = π ζ(3) − ζ(5) − π +
72 24 2160 5184
139 298 2 17101 2
+ ζ(3) − π ln(2) + π
18 9 810
100 1 ln4 (2) π 2 ln2 (2)
+ Li4 ( ) + − (8.24)
3 2 24 24
= 1.181 241 456 587 . . . . (8.25)
This remarkable analytic expression, easily evaluated to any desired numer-
ical precision with no theoretical error, is very signiﬁcant for determining
α from g/2 insofar as it completely removes what otherwise would be a
signiﬁcant numerical uncertainty.
Is the remarkable analytic expression free of mistakes? The best conﬁr-
mation is the good agreement between the extremely complicated analytic
derivation and a simpler but computation-intensive numerical calculation,
(6)
A1 = 1.181 259 (40) [44]. This result used the best computers available
many years ago; it could (and should) now be greatly improved. An earlier
numerical evaluation led to the discovery and correction of a mistake made
in an earlier analytic derivation of a renormalization term [44]. This further
illustrates the importance of checking analytic derivations numerically.
An exact analytic calculation of the 48 Feynman diagrams that deter-
(6)
mine the mass-dependent function A2 has also been completed [45, 46].
However, the resulting expressions are apparently too lengthy to publish in
a printed form. Instead, expansions for small mass ratios are made available
4
(6)
A2 (r) = r2k f2k (r). (8.26)
k=1
The expansions make it easy to calculate the two most important mass
dependent contributions to the precision at which the measurement uncer-
tainty in the mass ratios is important for any foreseeable improvements in
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
272 G. Gabrielse
the mass ratio uncertainties. Functions f2 and f4 are from Ref. 46, f6 is
from Refs. 45 and 47, and f8 is from Ref. 42.
23 ln(r) 3ζ(3) 2π 2 74957
f2 (r) = + − − , (8.27)
135 2 45 97200
4337 ln2 (r) 209891 ln(r) 1811ζ(3) 1919π 2
f4 (r) = − + + −
22680 476280 2304 68040
451205689
− , (8.28)
533433600
2807 ln2 (r) 665641 ln(r) 3077ζ(3)
f6 (r) = − + +
21600 2976750 5760
16967π 2 246800849221
− − , (8.29)
907200 480090240000
55163 ln2 (r) 24063509989 ln(r) 9289ζ(3)
f8 (r) = − + +
594000 172889640000 23040
340019π 2 896194260575549
− − . (8.30)
24948000 2396250410400000
These expansions have been compared to the exact calculations to verify the
claim that their accuracy is much higher than any experimental uncertainty
that will likely be reached [42].
With the current values of the mass ratios,
(6)
A2 (me /mµ ) = −7.373 941 58 (28) × 10−6 , (8.31)
(6) −8
A2 (me /mτ ) = −6.581 9 (19) × 10 . (8.32)
The uncertainties arise from the measurement imprecision in the mass ra-
tios, not from any theoretical uncertainty. The term that depends upon
both mass ratios [42],
(6)
A3 (me /mµ , me /mτ ) = 1.909 45 (62) × 10−13 , (8.33)
is too small to be important for the electron g/2 in the foreseeable future,
or to even have its uncertainty visible in Fig. 8.2..
For the current and foreseeable experimental precisions, only the mass-
independent term is required in eighth order. Kinoshita and his collabora-
tors have reduced the 891 Feynman diagrams to a much smaller number of
master integrals, which were then evaluated by Monte Carlo integrations
over the course of ten years. The latest result is [15]
(8)
C8 = A1 = −1.9144 (35), (8.34)
The uncertainty is that of the numerical integration as evaluated by an
integration routine [48], limited by the computer time available for the
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
Determining the Fine Structure Constant 273
integrations. A calculation of this coeﬃcient to 0.2% is a remarkable result
that is critical for determining α from g/2.
Checking the eighth order coeﬃcient to make sure that it is correctly
evaluated is a formidable challenge. There is no analytic result to compare
(yet). Only the collaborating groups of Kinoshita and Nio have had the
courage and tenacity needed to complete such a challenging calculation.
The complexity of the calculation makes is very diﬃcult to avoid mistakes.
The strategy has been to check each part of the calculation by using more
than one independent formulation [49].
Our 2006 measurement of g/2 came while the theoretical checking was
underway. At this point we published a value of α along with a warning
that the theoretical checking for eighth order was not yet complete [50]. In
2007, a calculation using an independent formulation reached a precision
suﬃcient to reveal a mistake [15] in how infrared divergences were handled
in two master integrals. When the mistake in C8 was corrected, the α
determined from g/2 shifted a bit [50].
One could take the moral of the 2007 adjustment to be that the sheer
complexity of the high order QED calculation makes it impossible to be cer-
tain that they are done correctly. I take the opposite conclusion, choosing
to be reassured that the theory is checked so carefully that even a very small
mishandling of divergences can be identiﬁed and corrected. Now that the
eighth order calculation is completely checked by an independent formula-
tion, to a level of precision that the theorists deem is suﬃcient to detect
mistakes, it seems much less likely that another substantial change in α
will be necessary. The check will be even better when the new calculation
reaches the numerical precision of the calculation being checked.
An evaluation of, or at least a reasonable bound on, the tenth order coef-
(10)
ﬁcient, C10 ≈ A1 , is needed as a result of the level of accuracy of our 2008
measurement of g/2. A calculation is not easy given that 12 672 Feynman
diagrams contribute. The estimated bound suggested in the meantime [37],
C10 = 0.0 (4.6), (8.35)
takes the uncalculated coeﬃcient to be zero with an uncertainty that is an
extrapolation of the size of the lower order coeﬃcients. This crude estimate
is not so convincing. It is especially unsatisfying given that it now limits
the accuracy with which α can be determined from the measured g/2, as
illustrated in Fig. 8.3..
8.3.3. How Much Better can α be Determined?
Fig. 8.3. shows the experimental and theoretical contributions to the uncer-
tainty in the α determined from g/2. This uncertainty is currently divided
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
274 G. Gabrielse
nearly equally between measurement uncertainty in g/2 and theoretical un-
certainty in the Standard Model relation between g/2 and α. The largest
theoretical uncertainty is from the uncalculated C10 , followed by numerical
uncertainty in C8 .
(10)
The ﬁrst calculation of C10 ≈ A1 is now underway [15, 51, 52]. It
has already produced an automated code that was checked by recomputing
the eighth order coeﬃcient. (This is the independent calculation that in
2007 reached the precision needed to expose a mistake in the calculation of
C8 [15].) No limit or bound will apparently be available until the impressive
calculation is completed at some level of precision because many contribu-
tions with similar magnitudes sum to make a smaller result. A completed
calculation of C10 will likely reduce the theoretical uncertainty enough so
that the uncertainty in α would approach the 0.26 ppb uncertainty that
comes from the measurement uncertainty in g/2.
The uncertainty in C8 can be reduced once the uncertainty in C10 has
been reduced enough to warrant this. More computation time would reduce
the numerical integration uncertainty in C8 . A better hope is that parts or
all of this coeﬃcient will eventually be calculated analytically. Eﬀorts in
this direction are underway [53].
It thus seems likely that the theoretical uncertainty that limits the accu-
racy to which α can be determined from g/2 can and will be reduced below
0.1 ppb. The corresponding good news is that it also seems likely that the
uncertainty in α from the measurement of g/2 can also be reduced below
0.1 ppb. With enough experimental and theoretical eﬀort it may well be
possible to do even better.
8.4. Determining α from the Rydberg, Two Mass Ratios and
/M for an Atom
All the determinations of α whose uncertainty is not much larger than 20
times the uncertainty of the α from g/2 are compared in Fig. 8.1.. The
values not from g/2 in this ﬁgure do not come from a single measurement.
Instead, each requires the determination of four quantities from a mini-
mum of six precise measurements, each measurement contributing to the
uncertainty in the α that is determined. Theory, including QED theory, is
essential to determining two of the measured quantities.
The deﬁnitions for α and the Rydberg constant R∞ taken together yield
4π
α2 = R∞ . (8.36)
c me
No accurate measurement of /me for the electron is available. However,
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
Determining the Fine Structure Constant 275
a precisely measured /Mx for a Cs or Rb atom (of mass Mx ) can be used
along with two measurable mass ratios, Ar (e) and Ar (x),
4π Ar (x)
α2 = R∞ . (8.37)
c Ar (e) Mx
The speed of light, c, is deﬁned in the SI system of units.
The ﬁrst of the needed mass ratios, Ar (e) = 12me /M (12 C), is the elec-
tron mass in atomic mass units (amu). The second is the mass of Cs or
Rb in amu, Ar (x) = 12M (x)/M (12 C). Determining the Rydberg constant
accurately requires the precise measurements of two hydrogen transition fre-
quencies (and less accurate measurements of other quantities). Determining
/Mx for Cs and Rb requires the measurement of an optical frequency ω
and an atom recoil velocity vr , or equivalent recoil frequency shift, ωr .
The fractional uncertainties that contribute to the uncertainty in α are
listed in Table 8.1. for Cs, and in Table 8.2. for Rb, in order of increasing
precision. Owing to the square in Eq. 8.37 the fractional uncertainty in α
is half the fractional uncertainty of the contributing measurements.
Table 8.1. Measurements determining α(Cs).
Measurement ∆α/α References
quantity ppb ppb
ωr 15. 7.7 [22]
Ar (e) 0.4 0.2 [23, 54]
Ar (Cs) 0.2 0.1 [18]
ω 0.007 0.007 [20]
R∞ 0.007 0.004 [16, 17, 23]
Best α(Cs) 8.0 [22]
Table 8.2. Measurements determining α(Rb)
Measurement ∆α/α References
quantity ppb ppb
ωr 9.1 4.6 [21]
Ar (e) 0.4 0.2 [23, 54]
Ar (Rb) 0.2 0.1 [18]
ω 0.4 0.4 [21]
R∞ 0.007 0.004 [16, 17, 23]
Best α(Rb) 4.6 [21]
The Rydberg constant describes the structure of a non-relativistic hy-
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
276 G. Gabrielse
drogen atom in the limit of an inﬁnite proton mass. Real hydrogen atoms, of
course, have ﬁne structure, Lamb shifts, and hyperﬁne structure. The pro-
ton has a ﬁnite mass. The Dirac energy eigenvalues must be corrected for
relativistic recoil, QED self-energy eﬀects, and QED vacuum polarization.
Corrections for nuclear polarization, nuclear size and nuclear self-energy are
important at the precision with which transition energies can be measured.
The theory needed to determine the Rydberg constant from measure-
ments is described in a seven page section of Ref. 23 entitled “Theory rele-
vant to the Rydberg constant.” The accepted value of the Rydberg comes
from a best ﬁt of the measurements of a number of accurately measured
hydrogen transitions [16, 17], the proton-to-electron mass ratio [19], the
size of the proton, etc. to the intricate hydrogen theory for each of the
hydrogen transitions, using more precisely measured values for every quan-
tity that is not determined best by ﬁtting. A full discussion of this process
and a complete bibliography for all the measurements and calculations that
make important contributions is beyond the scope of this work. The tables
thus show the currently accepted uncertainty for the Rydberg constant [23]
rather than the uncertainties from all the contributing measurements.
The measured electron mass in amu, Ar (e), relies equally upon precise
measurements [19, 54] and upon bound state QED theory [24], using
me gbound 1 ωc
= , (8.38)
M 2 q/e ωs
where q/e is the integer charge of the ion in terms of one quantum of
charge. Measurements are made using a 12 C 5+ (or 16 O7+ ) ion trapped in
a pair of open access Penning traps [55], a type of trap we developed for
accurate measurements of q/m for an antiproton. Spin ﬂips and cyclotron
excitations are made in one trap and then transferred to the other for
detection in a strong magnetic gradient. The spin frequency ωs of the
electron bound in an ion is measured. The cyclotron frequencies ωc of the
ion is deduced from the measurable oscillation frequencies of the trapped ion
using the Brown-Gabrielse invariance theorem [31, 56]. This determination
of the electron mass in amu could not take place without an extensive
QED calculation of the g value of an electron bound into an ion [24]. A less
accurate measurement of the electron mass in amu does not rely on QED
theory [57]; it agrees with the more accurate method.
The needed mass ratios, Ar (x), are from measurements [18] using iso-
lated ions in a orthogonalized hyperbolic Penning trap [58], a trap design
we developed to facilitate precise measurements. Ion cyclotron frequencies
are deduced from oscillation frequencies of the ions in the trap using the
same invariance theorem [31, 56]. Ion cyclotron energy is transferred to the
axial motion using a sideband method that allows cyclotron information to
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
Determining the Fine Structure Constant 277
be read out by a SQUID detector that is coupled to the axial motion of an
ion in a trap. Ratios of ion frequencies give the ratios of masses in a simple
and direct way that is insensitive to theory. Ratios to of Mx to the carbon
+
mass, as needed to get amu, came from using ions like CO2 and several
hydrocarbon ions as reference ions.
The basic idea of the /Mx measurements for Cs and Rb is that when
an atom absorbs a quantum of light from a laser ﬁeld, or is stimulated to
emit a quantum of light into a laser ﬁeld, then the atom recoils with a
momentum Mx vr = k, where for a laser ﬁeld with angular frequency ω we
have k = ω/c. Thus /Mx is determined by the measured optical frequency
of the laser radiation, ω, and by the atom recoil velocity vr . The latter can
be accurately measured from the recoil shift ωr in the resonance frequency
caused by the recoil of the atom.
The laser frequency is measured a bit diﬀerently for Cs and Rb. For Cs
the needed frequencies are measured with a precision of 0.007 ppb, much
more accurately than will likely needed for some time, using an optical
comb to directly measure the frequency with respect to hydrogen maser
and a Cs fountain clock [20]. For Rb, a diode laser is locked to a stable
cavity, and its frequency is compared using an intermediate reference laser
to that of a two-photon Rb standard [59].
The largest uncertainty in determining α using Eq. 8.37 is the uncer-
tainty in measuring the atom recoil velocity vr , or equivalently the recoil
shift ωr = 1 Mx vr / . This measurement uncertainty is much larger than
2
2
the measurement uncertainty in R∞ , Ar (e), Ar (x), and ω, and is thus
the limit to the accuracy with which α can currently be determined by
this method. The availability of extremely cold laser-cooled atoms has led
to signiﬁcant progress by two diﬀerent research groups. First came a Cs
measurement at Stanford [22] in 2002. More recently came 2006 and 2008
measurements of slightly higher precision with Rb atoms at the LKB in
Paris [21, 59].
The Cs recoil measurement [22] and the most accurate of the Rb mea-
surement [21] both measure the atom recoil using atom interferometry.
e
The so-called Ramsey-Bord´ spectrometer [60] conﬁguration that is used
in both cases was developed to apply Ramsey separated oscillator ﬁeld
methods at optical frequencies. Pairs of stimulated Raman π/2 pulses pro-
duced by counter propagating laser beams [61] split the wave packet of a
cold atom into two phase-coherent wave packets with diﬀerent atom veloci-
ties. A series of N Raman π pulses then add recoil kicks to both parts of the
atom wave packet. When a ﬁnal pair of Raman π/2 pulses make it possible
for the previously separated parts of the wave packet to interfere, the inter-
ference pattern reveals the energy diﬀerence, and hence the recoil frequency
diﬀerence, for the wave packets in the two arms of the interferometer.
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
278 G. Gabrielse
The measured phase diﬀerence that reveals vr and ωr goes basically
as N , where N is the number of additional recoil kicks given to the wave
packets in both arms of the spectrometer. The experiments diﬀer in the way
that they seek to make N as large as possible. The initial Cs measurement
used a sequence of π pulses to achieve N = 30. The most accurate of the
Rb measurements achieved N = 1600 using a series of Raman transitions
with the frequency diﬀerence between the counterpropagating laser beams
being swept linearly in time. This can equivalently be regarded as a type
of Bloch oscillation within an accelerating optical lattice [62].
An improved apparatus is under construction in the hope of improving
the 2008 measurement of the Rb recoil shift on the time scale of a year or
two. Although no Cs recoil measurement has been reported since 2002, an
improved apparatus has been built. A goal of soon measuring the Cs recoil
shift accurately enough to determine α to sub-ppb accuracy was mentioned
in a recent report on improved beam splitters for a Cs atom interferometer
[63].
8.5. Other Measurements to Determine α
8.5.1. Determining α from He Fine Structure
Surprisingly none of the accurate measurements determine α by measuring
atomic ﬁne structure intervals. Helium ﬁne structure intervals have been
measured precisely enough so that two-electron QED theory could deter-
mine α from the interval at about the same precision as do the combined
Rydberg, mass ratios and atom recoil measurements. Helium is a better
candidate for such measurements than is hydrogen because the ﬁne struc-
ture splittings are larger, and the radiation lifetimes of the levels are longer
so that narrow resonance lines can be measured. Unfortunately, theoretical
inconsistencies need to be resolved.
The most accurate measurements of three 23 P 4 He ﬁne structure inter-
vals [1–4, 25] are in good agreement as illustrated in Fig. 8.4.. Our Harvard
laser spectroscopy measurements [25] have the smallest uncertainties,
f12 = 2 291 175.59 ±0.51 kHz [220 ppb] (8.39)
f01 = 29 616 951.66 ±0.70 kHz [ 24 ppb] (8.40)
f02 = 31 908 126.78 ±0.94 kHz [ 29 ppb]. (8.41)
The ﬁgure shows good agreement between measurements of the largest
intervals; these are best for determining α. The measurements of the small
interval also agree well. This interval is less useful for determining α but is
a useful check on the theory.
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
Determining the Fine Structure Constant 279
(a) 2 3P 0
Harvard'05
f02 - f01
2 3P 1 Y ork'00
23P 2 T exas '00
LE NS '99
theory: W ars aw'06
theory: W inds or'02
140 150 160 170 180 190
frequency - 2 291 000 kHz
(b) 3
2 P0 Harvard'05 f02 - f12
3
2 P1 LE NS '04
3 Y ork'01
2 P2
T exas '00
theory: W ars aw'06
theory: W inds or'02
920 930 940 950 960
frequency - 29 616 000 kHz
(c) 2 3P 0 Harvard'05 f12 + f01
2 3P 1 Y ork'00-01
23P 2 T exas '00
LE NS '99
theory: W ars aw'06
theory: W inds or'02
90 100 110 120 130 140
frequency - 31 908 000 kHz
Fig. 8.4. Most accurate measured [1–4] and calculated [5, 6] 4 He ﬁne structure intervals
with standard deviations. Directly measured intervals (black ﬁlled circles) are compared
to indirect values (open circles) deduced from measurements of the other two intervals.
Uncorrelated errors are assumed for the indirect values for other groups.
Because a ﬁne structure interval frequency f goes as R∞ α2 to lowest
order, and the Rydberg is known much more accurately than α, a fractional
uncertainty in f translates into a fractional uncertainty for α that is smaller
by half – if the theory would contribute no additional uncertainty. The 24
ppb fractional uncertainty in the f01 that we reported back in 2005 would
then suﬃce to determine α to 12 ppb, a small enough uncertainty to allow
this value to be plotted with the most precise measurements in Fig. 8.1..
A big disappointment is that Fig. 8.4. reveals two serious problems with
calculations done independently by two diﬀerent groups [5, 6]. (More about
the calculations is in Chapter 7.) The calculated interval frequencies (using
α from g/2) are plotted below the measurements in the ﬁgure. The ﬁrst
problem is that the two calculations do not agree, raising questions as to
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
280 G. Gabrielse
whether mistakes have been made. It is not hard to imagine mistakes
given that the two-electron QED theory gives interval frequencies that are
the sum of a series in powers of both α and ln α. The convergence is not
rapid, and the many terms to be summed present a signiﬁcant bookkeeping
challenge. The second problem is that both theories disagree with the
measurements, for both the large and small intervals. The measurements
from 2005 and earlier, though they have an accuracy that would suﬃce to
be one of the most precise determinations of α, cannot be used until the
theory issues are resolved.
A serious diﬃculty with two-electron QED theory seems surprising given
how successful one-electron QED theory has been in its predictions. Is
there a fundamental problem or is this a case of mistakes? Until the two
calculations agree the latter explanation is hard to discount, and neither
calculation agrees with experiments.
A problem with the measurements is the other possibility, though the
good agreement between measurements with very diﬀerent systematic ef-
fects would suggest otherwise. One caution is that the most accurate mea-
surements determine to 700 Hz the center of resonance lines that are slightly
bigger than 1.6 MHz natural linewidths. “Splitting the line” to a few parts
in 104 of the linewidth is challenging, requiring as it does that systematic
shifts and distortions of the measured resonance lines be either insigniﬁ-
cant or well understood. It is hard to believe that a helium ﬁne structure
measurement could ever approach the accuracy of the current α from g/2.
After we published our measurement of the helium ﬁne structure inter-
vals we narrowed our laser linewidth to below 5 kHz and stabilized it to
an iodine clock using an optical comb that we built to bridge between the
very diﬀerent frequencies of our clock and the 1.08 µm optical transitions
that we measured. We also greatly improved the signal to noise ratio in our
measured resonance lines. Within a couple of hours we could get close to
100 Hz resolution for all three intervals, and we could do this in an auto-
mated way during the mechanically and electrically quiet night times with
none of us present.
However, at the new level of precision that we were exploring we encoun-
tered systematic frequency shifts that suggested to us that we had pushed
saturated absorption measurements in a discharge cell as far as they should
reasonably be pushed. Given the large amount of line splitting already be-
ing done, and the theoretical inconsistencies, we decided not to replace the
cell with a helium beam. Instead, several years ago we shut the experiment
down – perhaps the ﬁrst discontinued optical comb experiment – and de-
cided to pursue measurements of the electric dipole moment of the electron
instead.
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
Determining the Fine Structure Constant 281
8.5.2. Historically Important Methods
In Fig. 8.1. there is a factor of more than 20 between the sizes of the
uncertainties for the most accurate determinations of α that have already
been discussed above. All other measurement of α have larger error bars
that will not ﬁt on this scale. Several additional measurements ﬁt on the
8 times expanded scale of Fig. 8.5., though the error bars for the most
accurate determinations of α from g/2 are then too small to be visible.
ppb
100 50 0 50
Harvard g 2
UW g 2
Rb h m
Cs h m
quantum Hall
nhM
muonium hfs
Josephson
598.5 599.0 599.5 600.0 600.5 601.0
1 5
Α 137.03 10
Fig. 8.5. Less accurate measurements of α compared upon an expanded scale. The
uncertainties in the two most accurate determnations of α are too small to be visible on
this large scale.
A summary and discussion of traditional measurements of α is in Ref. 23.
The work includes the value deduced from the quantum Hall resistance [64],
a value that essentially agrees with the more accurate determinations of α
insofar as these lie almost within its one standard deviation error bars.
A measurement using neutrons [65] that is similar in spirit to the de-
scribed Cs and Rb measurements is also plotted. Diﬀerent mass ratios are
required, of course, but an even more important diﬀerence is that /Mn is
deduced from the diﬀraction of cold neutrons from a Si crystal. The lattice
spacing in Si is thus crucial, and there is an impressive range of diﬀering
values for this lattice constant [23]. A recommended value [23] is used for
the ﬁgure but given the range of measured lattice constants it is not so
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
282 G. Gabrielse
surprising that this value of α does not agree so well with more accurate
measurements.
Values from muonium hyperﬁne structure measurements [23, 43] and
from measurements of the AC Josephson eﬀect (with related measurements
[23]) are also plotted because of their importance in the past. It is not clear
why the latter solid state measurement disagree so much with the more
accurate values.
8.6. Conclusion
Combined measurements of the Rydberg constant, two mass ratios, a laser
frequency, and an atom recoil frequency together determine α using Cs
atoms to 8.0 ppb, and using Rb atoms to 4.6 ppb. Eﬀorts are underway to
improve both sets of measurements enough to determine α to 1 ppb.
Helium ﬁne structure measurements are now accurate enough to de-
termine α at nearly the same precision, but with completely diﬀerent sys-
tematic uncertainties. Unfortunately, the two-electron QED theory needed
to relate ﬁne structure intervals to α heeds to be clariﬁed before this can
happen.
New measurements of the electron magnetic moment g/2, along with
QED calculations, determine the ﬁne structure constant much more accu-
rately than ever before, to 0.4 ppb. The uncertainty in α will be reduced,
without the need for a more accurate measurement of g/2, when a ﬁrst
calculation of the tenth order QED coeﬃcient is completed. It seems rea-
sonable to reduce the experimental and theoretical contribution to determi-
nations of α from g/2 to 0.1 ppb or better in eﬀorts now underway, though
this will take some time.
Acknowledgements
Useful comments on this manuscript from F. Biraben, D. Hanneke, T.
Kinoshita, S. Laporta, P. Mohr, H. Mueller, M. Nio, M. Passera and E.
Remiddi are gratefully acknowledged. Support for this work came from the
NSF, the AFOSR, and from the Humboldt Foundation.
Bibliography
[1] G. Giusfredi, P. de Natale, D. Mazzotti, P. C. Pastor, C. de Mauro, L. Fal-
lani, G. Hagel, V. Krachmalnicoﬀ, and M. Inguscio, Can. J. Phys. 83,
301–309, (2005).
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
Determining the Fine Structure Constant 283
[2] J. Castillega, D. Livingston, A. Sanders, and D. Shiner, Phys. Rev. Lett. 84,
4321–4324, (2000).
[3] C. H. Storry, M. C. George, and E. A. Hessels, Phys. Rev. Lett. 84, 3274–
3277, (2000).
[4] M. C. George, L. D. Lombardi, and E. A. Hessels, Phys. Rev. Lett. 87,
173002, (2001).
[5] G. W. F. Drake, Can. J. Phys. 80, 1195–1212, (2002).
[6] K. Pachucki, Phys. Rev. Lett. 97, 013002, (2006).
[7] D. Hanneke, S. Fogwell, and G. Gabrielse, Phys. Rev. Lett. 100, 120801,
(2008).
[8] B. Odom, D. Hanneke, B. D’Urso, and G. Gabrielse, Phys. Rev. Lett. 97,
030801, (2006).
[9] R. S. Van Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt, Phys. Rev. Lett.
59, 26–29, (1987).
[10] J. Schwinger, Phys. Rev. 73, 416L, (1948).
[11] C. M. Sommerﬁeld, Phys. Rev. 107, 328, (1957).
[12] A. Petermann, Helv. Phys. Acta. 30, 407, (1957).
[13] C. M. Sommerﬁeld, Ann. Phys. (N.Y.). 5, 26, (1958).
[14] S. Laporta and E. Remiddi, Phys. Lett. B. 379, 283, (1996).
[15] T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, Phys. Rev. Lett. 99,
110406, (2007).
[16] T. Udem, A. Huber, B. Gross, J. Reichert, M. Prevedelli, M. Weitz, and
a
T. W. H¨nsch, Phys. Rev. Lett. 79, 2646–2649, (1997).
[17] C. Schwob, L. Jozefowski, B. de Beauvoir, L. Hilico, F. Nez, L. Julien,
F. Biraben, O. Acef, J. J. Zondy, and A. Clairon, Phys. Rev. Lett. 82,
4960–4963, (1999).
[18] M. P. Bradley, J. V. Porto, S. Rainville, J. K. Thompson, and D. E.
Pritchard, Phys. Rev. Lett. 83, 4510–4513, (1999).
a
[19] G. Werth, J. Alonso, T. Beier, K. Blaum, S. Djekic, H. H¨ﬀner, N. Her-
u
manspahn, W. Quint, S. Stahl, J. Verd´, T. Valenzuela, and M. Vogel, Int.
J. Mass Spectrom. 251, 152, (2006).
[20] V. Gerginov, K. Calkins, C. E. Tanner, J. J. McFerran, S. Diddams, A. Bar-
tels, and L. Hollberg, Phys. Rev. A. 73, 032504, (2006).
e e
[21] M. Cadoret, E. de Mirandes, P. Clad´, S. Guellati-Hk´lifa, C. Schwob,
F. Nez, L. Julien, and F. Biraben, Phys. Rev. Lett. 101, 230801, (2008).
[22] A. Wicht, J. M. Hensley, E. Sarajlic, and S. Chu, Phys. Scr. T102, 82–88,
(2002).
[23] P. J. Mohr, B. N. Taylor, and D. B. Newall, Rev. Mod. Phys. 80, 633, (2008).
[24] K. Pachucki, A. Czarnecki, U. Jentschura, and V. A. Yerokhin, Phys. Rev.
A. 72, 022108, (2005).
[25] T. Zelevinsky, D. Farkas, and G. Gabrielse, Phys. Rev. Lett. 95, 203001,
(2005).
[26] G. Giusfredi, P. de Natale, D. Mazzotti, P. C. Pastor, C. de Mauro, L. Fal-
lani, G. Hagel, V. Krachmalnicoﬀ, and M. Inguscio, Can. J. Phys. 83,
301–309, (2005).
[27] I. M. Mills, P. J. Mohr, T. J. Quinn, B. N. Taylor, and E. R. Williams,
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
284 G. Gabrielse
Metrologia. 43, 227, (2006).
[28] G. W. Bennett and et al., Phys. Rev. D. 73, 072003, (2006).
[29] S. Peil and G. Gabrielse, Phys. Rev. Lett. 83, 1287–1290, (1999).
[30] G. Gabrielse and F. C. MacKintosh, Intl. J. of Mass Spec. and Ion Proc.
57, 1–17, (1984).
[31] L. S. Brown and G. Gabrielse, Phys. Rev. A. 25, 2423–2425, (1982).
[32] G. Gabrielse and H. Dehmelt, Phys. Rev. Lett. 55, 67–70, (1985).
[33] J. Tan and G. Gabrielse, Phys. Rev. Lett. 67, 3090–3093, (1991).
[34] B. D’Urso, R. Van Handel, B. Odom, D. Hanneke, and G. Gabrielse, Phys.
Rev. Lett. 94, 113002, (2005).
[35] B. D’Urso, B. Odom, and G. Gabrielse, Phys. Rev. Lett. 90(4), 043001,
(2003).
[36] P. J. Mohr and B. N. Taylor, Rev. Mod. Phys. 72, 351–495, (2000).
[37] P. J. Mohr and B. N. Taylor, Rev. Mod. Phys. 77, 1 – 107, (2005).
[38] A. Czarnecki, B. Krause, and W. J. Marciano, Phys. Rev. Lett. 76, 3267 –
3270, (1996).
[39] A. Petermann, Nucl. Phys. 5, 677, (1958).
[40] R. Karplus and N. M. Kroll, Phys. Rev. 77, 536, (1950).
[41] H. H. Elend, Phys. Rev. Lett. 20, 682, (1966). 21, 720(E) (1966).
[42] M. Passera, Phys. Rev. D. 75, 013002, (2007).
[43] W. Liu, M. G. Boshier, O. v. D. S. Dhawan, P. Egan, X. Fei, M. G.
Perdekamp, V. W. Hughes, M. Janousch, K. Jungmann, D. Kawall, F. G.
Mariam, C. Pillai, R. Prigl, G. z. Putlitz, I. Reinhard, W. Schwarz, P. A.
Thompson, and K. A. Woodle, Phys. Rev. Lett. 82, 711–714, (1999).
[44] T. Kinoshita, Phys. Rev. Lett. 75, 4728, (1995).
[45] S. Laporta, Nuovo Cim. A. 106A, 675 – 683, (1993).
[46] S. Laporta and E. Remiddi, Phys. Lett. B. 301, 440 – 446, (1993).
u
[47] J. H. K¨hn, et al., Phys. Rev. D. 68, 033018, (2003).
[48] G. P. Lepage, J. Comput. Phys. 27, 192 – 203, (1978).
[49] T. Kinoshita and M. Nio, Phys. Rev. Lett. 90, 021803, (2003).
[50] G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, and B. Odom, Phys. Rev.
Lett. 97, 030802, (2006). ibid. 99, 039902 (2007).
[51] T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, Nucl. Phys. B740,
138, (2006).
[52] T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, Phys. Rev. D. 78,
113006, (2008).
[53] S. Laporta and E. Remiddi. (private communication).
a
[54] T. Beier, H. H¨ﬀner, N. Hermanspahn, S. G. Karshenboim, H.-J. Kluge,
u
W. Quint, S. Stahl, J. Verd´, and G. Werth, Phys. Rev. Lett. 88, 011603,
(2002).
[55] G. Gabrielse, L. Haarsma, and S. L. Rolston, Intl. J. of Mass Spec. and Ion
Proc. 88, 319–332, (1989). ibid. 93, 121 1989.
[56] G. Gabrielse, Int. J. Mass Spectrom. 279, 107, (2009).
[57] D. L. Farnham, R. S. Van Dyck, Jr., and P. B. Schwinberg, Phys. Rev. Lett.
75, 3598–3601, (1995).
[58] G. Gabrielse, Phys. Rev. A. 27, 2277–2290, (1983).
March 28, 2009 8:26 World Scientiﬁc Review Volume - 9in x 6in leptmom
Determining the Fine Structure Constant 285
e e
[59] P. Clad´, E. de Mirandes, M. Cadoret, S. Guellati-Kh´lifa, C. Schwob,
F. Nez, L. Julien, and F. Biraben, Phys. Rev. Lett. 96, 033001, (2006).
Phys. Rev. A 74, 052109 (2006).
e
[60] C. Bord´, Phys. Lett. A. 140, 10, (1989).
[61] D. S. Weiss, B. C. Young, and S. Chu, Phys. Rev. Lett. 70, 2706–2709,
(1993).
[62] E. Peik, M. B. Dahan, I. Bouchoule, Y. Castin, and C. Salomon, Phys. Rev.
D. 55, 2289, (1997).
u
[63] H. M¨ller, S. Chiow, Q. Long, S. Herrmann, and S. Chu, Phys. Rev. Lett.
100, 180405, (2008).
[64] A. M. Jeﬀery, R. E. Elmquist, L. H. Lee, J. Q. Shields, and R. F. Dziuba,
IEEE Trans. Instrum. Meas. 46, 264, (1997).
u
[65] E. Kr¨ger, W. Nistler, and W. Weirauch, Metrologia. 36(2), 147–148, (1999).